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Region 1: When the load \(\rho ^{(n)}\)  stays constant (\(\alpha =0\) ) and the server needs also stay constant (\(\gamma =0\) ), does the queueing probability, \(P^{(n)}_Q\) , diminish as \(n\rightarrow \infty \) ?
Our result: We show that \(P^{(n)}_Q\)  is indeed diminishing in this region. This result should not be surprising given the classical result that when jobs have unit server need, the queueing probability is diminishing under constant load (see, e.g., [1]}, where a special case of their results analyzes a system where each job has a server need of one).

Region 2: When the load \(\rho ^{(n)}\)  stays constant (\(\alpha =0\) ) but the server needs scale (\(\gamma >0\) ), does the queueing probability, \(P^{(n)}_Q\) , diminish as \(n\rightarrow \infty \) ?
Our result: We show that \(P^{(n)}_Q\)  is diminishing when \(\gamma <1\) , i.e., when a job drawn from the class of highest server need occupies a diminishing fraction of the servers during service. The threshold \(\gamma <1\)  is tight in the sense that for a single job-class system, \(P^{(n)}_Q\)  is not diminishing when \(\gamma =1\) . This is easy to see by noting that when \(\gamma =1\) , the single job-class system is equivalent to a classical M/M/\(s\)  system where \(s=\lfloor n/m^{(n)}_{\max }\rfloor \)  is a constant and the load is also a constant.

Region 3: When the load \(\rho ^{(n)}\)  is in heavy-traffic (\(\alpha >0\) ) and the server needs stay constant (\(\gamma =0\) ), does the queueing probability, \(P^{(n)}_Q\) , diminish as \(n\rightarrow \infty \) ?
Our result: We show that \(P^{(n)}_Q\)  is diminishing when the load satisfies \(0<\alpha <1/2\) , which is a traffic regime analogous to the sub-Halfin-Whitt regime in the literature. The threshold \(\alpha <1/2\)  is tight in the sense that for a single job-class system, \(P^{(n)}_Q\)  is not diminishing when \(\alpha \ge 1/2\) . Again, this can been seen by noting that when \(\alpha \ge 1/2\) , the single job-class system is equivalent to a classical M/M/\(s\)  system with \(s=\lfloor n/m^{(n)}_{\max }\rfloor =\Theta (n)\) , whose traffic is at least as heavy as the Halfin-Whitt regime.

Region 4: When the load \(\rho ^{(n)}\)  is in heavy-traffic (\(\alpha >0\) ) and the server needs also scale (\(\gamma >0\) ), does the queueing probability, \(P^{(n)}_Q\) , diminish as \(n\rightarrow \infty \) ? This is the most interesting region since this is when the load and the server needs jointly scale.
Our result: We show that \(P^{(n)}_Q\)  is diminishing when \(2\alpha +\gamma <1\) . This exact characterization of the joint scaling regime for diminishing queueing probability formalizes the intuition that queueing is negligible in large systems if each job does not need too many servers and the load is not too heavy.
This threshold \(2\alpha +\gamma <1\)  is also tight in the sense that for a single job-class system, the queueing probability is not diminishing when \(2\alpha +\gamma \ge 1\) . To see this, we again consider the equivalent M/M/\(s\)  system with \(s=\lfloor n/m^{(n)}_{\max }\rfloor \) . It is not hard to verify that the load of this equivalent system is \(1-\Theta (n^{-\alpha })=1-\Theta (s^{-\frac{\alpha }{1-\gamma }})\) , which is at least as heavy as Halfin-Whitt since \(\frac{\alpha }{1-\gamma }\ge 1/2\)  under the condition that \(2\alpha +\gamma \ge 1\) .

